TheoryoftheFirm

Monopolis2cMarkets

Monopolyvs.Compe22on

•  Manyprice-takingfirms

•  Freeentry/exit

PerfectCompe,,on

•  Onefirm(marketpower)

•  Barrierstoentry

Monopoly

Monopolyvs.Compe22on

Monopoly

PerfectCompe,,on

Q

P

D

Monopoly’s demand = Market demand (ΔQ ⇒ ∇P)

q

P

P d

Firm’s demand = Horizontal line ( Δq ⇒ P does not change)

CausesofMonopoly

•  Natural1.Scarcityofaresource:onefirmownsakeyresource

(DeBeers’Diamonds)

2.Technologicalreasons:existenceoflargeeconomiesofscale

rela2vetothemarketsize--onefirmmayservethedemand

moreefficientlythanseveralfirms(water,telephone,…)

•  Legal1.Licenses(taxis,pharmacies,notaries,…)

2.Patents(medicaldrugs,intellectualproperty,…)

NaturalMonopoly

AC

MC

D

MR Q QM

P

PM

Anaturalmonopolyariseswhentherealargeeconomiesof

scale:costminimiza2onwouldleadtoasinglefirmproducing

thetotaloutput.

TheMonopolist’sProblem

Therevenuefunc2onofthemonopolyis

R(q)=p(q)q,

wherep(q)istheINVERSEdemandfunc2on.

Thereforetheproblemofthemonopolis2cfirmis

Maxq≥0

π(q)=p(q)q–C(q).

ProfitMaximiza2on

•  FirstOrderCondi2on:

•  SecondOrderCondi2on:

•  ShutdownCondi2on:π(qM)≥π(0)

€

π '(q) = 0⇔ MR(q) = MC(q).

0)(')('')('2 ≤−+ qMCqqpqp

ProfitMaximiza2on

R(q)

C(q)

π(q)

Slope=MR

Slope=MC

q qM

C,R, π

ProfitMaximiza2on:MR

Marginalrevenueisgivenby

Thetwotermsinthisexpressionhaveaveryclearinterpreta2on:

thesaleofoneaddi2onal(infinitesimal)unitgenerates

(a)  anincreaseofrevenueequaltotheprice(quan2tyeffect),and(b)  adecreaseofrevenueequaltothetotaloutput2mesthe

reduc2oninpricenecessarytogeneratethedemandofthis

addi2onalunit(priceeffect).

( ) .)(')()()()((+) −

+== qqpqpqqpdqdqMR

ProfitMaximiza2on:MR

P

q

P(q)

A:quan2tyeffect

B:priceeffect

MR=A–B.

B

A

q q +1

ProfitMaximiza2on:MR

q

p

MR(q)

MarginalRevenue:

MR(q)=p(q)+p’(q)q.

(Sincep’(q)<0,theMR

curveisalwaysbelow

thedemandcurve.)

P(q)

ProfitMaximiza2on:SOC

TheSecondOrdenCondi2onis:

Sincep’(q)<0,thecondi2onC’’(q)>0(requiringthatthe

firmhasdiseconomiesofscaleneartheop2mallevelof

output)isnolongerrequiredforprofitmaximiza2on:at

theop2malproduc2onlevelthefirmmayhaveeconomies

ofscale.

0')('')('2 ≤−+ MCqqpqp

MonopolyEquilibrium

Monopoly PerfectCompe22on

AC

MC

DMR

q

p

pM

qM

CS

PS

DWL

p

q

D

MC

ACCS

PS

qC

pC

MonopolyEquilibrium

Inamonopoly,thepriceisabovethecompe22veprice,

pM>p

C,

theoutputisbelowthecompe22veoutput,

qM<q

C,

theconsumer(producer)surplusisbelow(above)thecompe22vesurplus

CSM<CS

C(PS

M>PS

C),

andthetotalsurplusisbelowthecompe22vesurplus,

TSM<TS

C.

Also,thereisaDeadweightLoss,DWL=TSC-TS

M>0.

SurplusandProfits

qM

pM

DMR

MC=

ACpC

qC

q

p

π=PS

CSDWL

LernerIndex

FirstOrderCondi2onforprofitmaximiza2onis:

MR(q)=MC(q).

Marginalrevenueis

).(')(')( qppqppqqppqMR ⎟⎟⎠

⎞⎜⎜⎝

⎛+=+=

LernerIndex

Wecanwritethemonopolypriceasafunc2onofthe

demandprice-elas2city.Thedemandprice-elas2cityis

Then,

€

ε = q'(p) pq

=p

qp'(q).

€

MR = MC⇔ p 1+1ε

⎛

⎝ ⎜

⎞

⎠ ⎟ = MC.

LernerIndex

Ifdemandisveryelas2c(Ɛhigh),themarginwillbesmall;

andviceversa:

pM p

M

MC MC

p p

qM q

Mq q

MC(qM)

MC(qM)

MR

MR

DD

LernerIndex

TheLernerindexmeasuresmonopolypower:itexplainsthe

percentageofpricethatisnotduetocosts.

Itisdefinedas

Notethat0≤L≤1.(L=0underperfectcompe22on.)

€

L =pM −MC(qM )

pM= −

1ε.

Example

Amonopolistfacesthedemandfunc2on

D(p)=max{12–p,0},anditscostfunc2onis

C(q)=5+4q.

Wecalculatethemonopolyequilibrium,theconsumerand

producersurplusesandthedeadweightloss,themonopoly

profits,theLernerindex.

Example

MonopolyEquilibrium.Revenueis

R(q)=(12–q)q=12q–q2.

Hence

MR(q)=12–2q.

SinceMC(q)=4,theequilibriumoutputisgivenbytheequa2on

MR(q)=MC(q)⇒12–2q=4.

Thus,qM=4,andp

M=12–q

M=12–4=8.

Example

Wecalculatestheconsumerandproducersurplusandthe

monopoly’sprofit:

CS=0.5*(12–pM)*q

M=0.5*(12-8)*4=8;

PS=(pM–CMa)*q

M=(8-4)*4=16;

π(q)=pMqM–C(q

M)=8*4–(5+4*4)=11;

Sincethecompe22veequilibriumispC=4,q

C=8,wehave:

DWL=0.5*(pM–p

C)(q

C–q

M)=0.5*4*4=8.

Example

WecalculatetheLernerindex:

LI=(pM–MC(q

M))/p

M=(8-4)/8=0.5

12

4

4 8 12

8

p

q

MC

MR

D

CS

PSDWL

Monopoly

PerfectCompe22on

Regula2ngaMonopoly

Anunregulatedmonopolygeneratesadeadweightloss,DWL>0.

Canthisbeavoided?

Anobvioussolu2on(ifpossible–thatis,ifweknowthe

monopoly’scostfunc2on),istoimposearegulatedpriceequal

tomarginalcost,thusgenera2ngthecompe22veoutcome.

Thissolu2onmayleadtomonopoly’slossesthatwouldhaveto

besubsidized.

Whensubsidizingisnotpossibleorconvenient(risingthe

necessaryrevenuemayinvolveotherinefficiencies),an

alterna2veregulatedpricemaybetheaveragecost.

D

AC

MC

MR

p

q

Subsidy=qC*(AC(q

C)–p

C)

qC

pC

AC(qC)

P=MC(q)

Regula2ngaMonopoly

p

q

D

AC

MC

MR

P=AC(q)

qC

qR

pC

pR

Regula2ngaMonopoly

Deadweithloss

Taxes

Inamonopoly,asinacompe22vemarket,introducing

ataxcausesanincreaseinpricesandadecreasein

output,andtherefore,adeadweightloss.

Thepropor2onofthetaxthatispaidforbyconsumers

depends,inamonopolyaswellasinacompe22ve

market,ofthepriceelas2cityofthedemand.

PriceCaps

Recallthatinacompe22vemarketintroducingapricecap(i.e.,a

maximumpriceatwhichthegoodmaybetraded),reducesthe

levelofoutput(andcreatesanexcessdemandthatrequires

ra2oningthedemand),resultsinadeadweightloss(reducesthe

totalsurplus)andmayormaynotincreasetheconsumer

surplus.

Inamonopoly,however,introducingapricecapbelowtheequilibriumpricemaybeaneffec2veinstrumenttodecreasethe

priceandincreasetheoutput.Moreover,ifthepricecapis

abovemarginalcost,thenitdoesnotcreateanexcessdemand,

andresultsinagreaterconsumerandtotalsurplus,andsmaller

deadweightloss.

TaxesandPriceCap:anExample

Consideramonopolistwhosecostfunc2onis

C(q)=5+4qfacingthedemand

D(p)=max{12–p,0}.Themonopolyoutputisthesolu2ontotheequa2on

12-2q=4

ThereforeqM=4,andp

M=8.

1.  SupposeweintroduceataxT=1€/unit.

Demand: D(p,T)=max{12–(p+T),0}.Revenue: R(q,T)=(12–q–T)q=12q–q2–Tq.

MarginalRevenue: MR(q)=12–2q–T.

Monopoly:Taxes

Theequa2onMR(q)=MC(q)isnow

12–2q–T=4

HenceoutputisqM(T)=4–T/2,theconsumerpriceis

pM(T)=8+T/2,and

monopolyprice=8+T/2–T=8–T/2.

Thatis,thetaxburdenissharedequallybetweentheconsumers

andthemonopoly.

Monopoly:Taxes

D(p)D(p,T)

MC

qM

qT

pM

pC

pS

T

p

q

12

11

4

Monopoly:Taxes

MR(q)=MC(q)⇒12–2q–T=4⇒q(T)=4–T/2.

WiththetaxeT,thepricepaidbytheconsumerandtheprice

receivedbymonopolyarenotthesame:

Pricepaidbyconsumer=12–q(T)=8+T/2

Pricereceivedbymonopoly=8+T/2–T=8–T/2.

Inthiscase,thetaxburdenissharedequallybetweenthe

consumersandthemonopoly.

TaxesandPriceCap:anExample

2.Supposeweintroduceapricecapp+<8=pM.

Thepricecapchangesderevenuefunc2onofthemonopoly

TheMarginalRevenueis:

Parap+=7,theequilibriumisq=5andp=p+=7(seethegraph

inthepage).

€

R(p+,q) =12 − q if 12 - q ≤ p+

p+q if 12 − q > p+

⎧ ⎨ ⎩

⎫ ⎬ ⎭

€

MR(p+,q) =12 − 2q if 12 - q ≤ p+

p+ if 12 − q > p+

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Monopoly:Taxes

D(p)

MR+

MC

q+=5

pM=8

pC

p+=7

p

q

12

11

4

qM=4

PriceDiscrimina2on

Sofar,wehaveassumedthatthemonopolycharges

thesamepriceforeachunit.

Wenowdiscusshowthemonopolymayincreaseits

profitbychargingdifferentpricestodifferent

consumerswithdifferentWILLINGNESSTOPAY.

PriceDiscrimina2on

•  Firstdegree:themonopolysellseachinfinitesimalunitatthe

maximumpriceaconsumeriswillingtopay.Withthispricing

policyweobtain

CS=0,PS=TS=maximumsurplus,DWL=0.

•  Seconddegree:ThemonopolyusesNON-LINEARpricing

policies:forexample,volumediscounts.Thistypeof

discrimina2onisverycommoninwater,electricity,telephone

andinternetsupplies.

ThirdDegreePriceDiscrimina2on

Themonopolytriestosegmentthemarket(by

geographicalcriteria,bycustomers’features,etc.),and

chargesdifferentpricestoeach“marketsegment.”

Example:Therearetwogroupsofconsumers,1and2,

whosedemandsareD1(p

1)andD

2(p

2).

ThirdDegreePriceDiscrimina2on

Ɛ2>Ɛ

1

Group1 Group2p1 p2

q2q1

D1

MR1

MR2

D2

ThirdDegreePriceDiscrimina2on

Themonopolistchoosesq1andq

2tomaximizethetotal

profits:

π(q1,q

2)=R

1(q

1)+R

2(q

2)–C(q

1+q

2)

=p1(q

1)q

1+p

2(q

2)q

2–C(q

1+q

2)

FOC:

).()()()(

2122

2111

qqMCqMRqqMCqMR+=

+=

Intermsofelas2ci2es:

p1(1+1/Ɛ

1)=p

2(1+1/Ɛ

2)=MC(q

1+q

2)

Therefore

p1/p

2=(1+1/Ɛ

2)/(1+1/Ɛ

1)<1;

and

p1<p

2;

Thatis,monopolypriceislowerinthemarketwithamore

elas2cdemand.

ThirdDegreeDiscrimina2on

Supposeamonopolyproducesagoodwithcosts

C(q)=q2/2,

andfacestwomarketswithdemands

D1(p

1)=max{20–p

1,0}

and

D2(p

2)=max{60–2p

2,0}.

Determinethequan22es,pricesandtotalprofitsunder

thirddegreepricediscrimina2on,andwithoutprice

discrimina2on.

ThirdDegreePriceDiscrimina2on:

Example

(a)Thirddegreediscrimina2on:

MC(q1+q

2)=q

1+q

2

R1=20q

1–q

12⇒MR

1=20–2q

1

R2=30q

2–0.5*q

22⇒MR

2=30–q

2

Equilibrium:

20–2q1=30–q

2=q

1+q

2

Solving,weobtainq1=2,q

2=14,p

1=18andp

2=23.

Monopolyprofitsareπ=18*2+14*23–C(2+14)=230.

ThirdDegreePriceDiscrimina2on:

Example

(b)Withoutpricediscrimina2on:theaggregatedemandis

Monopolyequilibrium:MR(q)=MC(q)

Solving,weobtainqM=15,p

M=22.5.

Monopolyprofitsare:π=84.375.

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

<−

≤≤−

=⇒⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

>

≤<−

≤−

=

80q if 0

20q if 2

30

80q20 if 3

80

)(30p if 0

30p20 if 26020p if 380

)( q

q

qppp

pD

ThirdDegreePriceDiscrimina2on:

Example